# Full text of "The Bernouilli Equation and the Torricelli Apparatus"

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```The Bernoulli Equation

Patrick Bruskiewich, M.Sc.

Department of Physics and Astronomy,
University of British Columbia
Vancouver, B.C., Canada V6T 1Z1

A

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4,

Bernoulli's Principle: where the velocity of an
incompressible fluid (gas or liquid) is high, the
pressure is low and where the velocity is low,
the pressure is high.

Bernoulli's Principle makes sense: Fluids will
flow from a region of higher pressure to a region
of lower pressure and in doing so increase their
speed as a result.

If this were not the case => if the pressure were
higher at a point of higher velocity then the
fluid would be slowed down at that point. As
a result this would be a point of lower velocity.
This is a contradiction!

?

Bernouilli developed an equation of state that
expresses his principle quantitatively. Assume
the fluid flow is steady and incompressible.

fe£ " v

Volume ~V,

V^tavwe r V*

f77777^'

Consider the work to move a small volume V\
of fluid at point 1,

W 1 = F x Ah = P 1 A 1 Al 1

(1)

At point 2 the force exerted on the fluid is
opposite to the motion so then

W 2 = - F 2 Al 2 = - P 2 A 2 Al 2

(2)

3

Work is also done on the fluid to counter the
force of gravity.

The work done to lift the mass of fluid in volume
Vi from point 1 to point 2 is

W g = - mg(y 2 - y\) (3)

So then the net work W net done on the fluid in
moving it from point 1 to point 2 is and is given
by

W net = W 1 + W 2 + Wg

= P 1 A 1 Al 1 - P 2 A 2 Al 2 - mg(y 2 -yi) (4)

According to the work-energy principle, the net
work on a system is equal to its change in kinetic
energy. That is

1 1

W net = ARE — -mv 2 - -mvi (5)

So then

1 9 1 o

-rnv 2 - -mv 1 = P 1 A 1 Al 1 - P 2 A 2 Al 2 - mg(y 2 - Vi)

(6)

Gather the 1 and 2 terms together we find

1 2

P 2 A 2 Al 2 + -mv 2 + mgy 2

i t~\

PlA 1 Ali + - mv l + m 9V\

Z

R

We can now factor out the volume on each
side of the equation, remembering that p = y t
namely

1 2
A 2 AZ 2 (P 2 + ~pv 2 + P9V2)

1 2

= AiAZi(Pi + -pv 1 + P^yi)

(7)

We know from the continuity equation that A 2 AZ 2
= AiAZi, which is the volume of the fluid ele-
ment flowing through the system.

We can therefore cancel these terms off both
sides of the equation.

What we are left with is

1 2

P 2 + ~pv 2 + P9V2
Pi + ~pv{ + pgy ±

This is Bernoulli's equation

Since points 1 and 2 can be any points along a
tube of flow, Bernouilli's equation can be writ-
ten

1 2

p 2 + ~P v 2 + P 9 V2 = constant (8)

a.

at every point in the fluid where y is the height
of the centre of the tube above the reference
level.

The Torrjcelli Apparatus

b* %-y t

v;*o

%x -r

r,

AX

H

The top of the apparatus and the spigot are
open to the atmosphere so that J\ = P 2 =

*atm ■

Using Bernoulli's equation (> 2 « 0) we have

~P v i + P 9 Vi = P 9 V2

(9)

Solving for v\ we find

,2 _

vf = 2g(v2 - vi) (io)

so that

vi = \][2g(y 2 - vi)] (11)

This is known as Torricelli's theorem, which was
discovered about 100 years before the Bernoulli
equation was developed.

Torricelli was a student of Galileo Galilei.

Experiment using Torricelli's apparatus

Objective: measure the distance the flow goes
before hitting the ground (Ax) and relate this
to the height of the fluid column (ft = y^ — 2/1).

Data points for Ax and h are collected and a
best fit power law curve is graphed. Assume
that Ax and h are related by a power law given
by

V

Ax = a ft]

n

(12)

h

where n is to be determined.

By taking the natural logarithm of the data
points and graphing the relationship we should
find that the slope of the graph is the power of
the relationship. That is

U[\$

X

u[W

A [In Ax]
A [In ft]

= n = slope

(13)

10

Theoretical Prediction

We know that

Ax

v = (14)

At V J

The distance m is related to the time it takes
the drop to fall by 2/1 = §#t 2 , so then

v = Ax

9

2j/i

= \/[2 £ h] (15)

where h = y 2 — 2/1 ■ This means that in theory

Ax = 2[yi h]2 (16)

where 2/1 is the height of the spigot above the
ground. Theory therefore tells us that the slope
of the datapoints on the log-log Ax and h graph
should be equal to i

1 1

```